## Balanced Sculptures

### Problem 275

Published on Friday, 22nd January 2010, 05:00 pm; Solved by 411; Difficulty rating: 85%Let us define a *balanced sculpture* of order `n` as follows:

- A polyomino made up of
`n`+1 tiles known as the*blocks*(`n`tiles)

and the*plinth*(remaining tile); - the plinth has its centre at position (
`x`= 0,`y`= 0); - the blocks have
`y`-coordinates greater than zero (so the plinth is the unique lowest tile); - the centre of mass of all the blocks, combined, has
`x`-coordinate equal to zero.

When counting the sculptures, any arrangements which are simply reflections about the `y`-axis, are __not__ counted as distinct. For example, the 18 balanced sculptures of order 6 are shown below; note that each pair of mirror images (about the `y`-axis) is counted as one sculpture:

There are 964 balanced sculptures of order 10 and 360505 of order 15.

How many balanced sculptures are there of order 18?